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Maximal injective subalgebras in factors associated with free groups. (English) Zbl 0545.46041
In this paper concrete examples of maximal injective von Neumann algebras in type $$II_ 1$$ factors are constructed. In particular if the free group of n generators $${\mathbb{F}}_ n$$ acts freely on some nonatomic probability space $$(X,\mu)$$ by measure preserving automorphisms and M denotes the associated group measure algebra and $$R_ u$$ denotes the injective subalgebra of M corresponding to the action of the generator $$u\in {\mathbb{F}}_ n$$ on $$(X,\mu)$$, then $$R_ u$$ is a maximal injective von Neumann subalgebra of M. The subalgebra $$R_ u$$ can be any injective type $$II_ 1$$ von Neumann algebra provided with a suitable action of $${\mathbb{F}}_ n$$ on $$(X,\mu)$$.
Two old problems of R. V. Kadison on the embedding of the hyperfinite factor $$R_ u$$ posed in the Baton Rouge Conference are also solved.
Reviewer: V.I.Ovchinnikov

##### MSC:
 46L10 General theory of von Neumann algebras 46L35 Classifications of $$C^*$$-algebras 22D15 Group algebras of locally compact groups 46L55 Noncommutative dynamical systems
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